math.sin on complex, imaginary part

Percentage Accurate: 54.6% → 99.0%

Time: 9.7s

Alternatives: 10

Speedup: 2.8×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (log1p (expm1 (* im (* -2.0 (cos re)))))))

C

double code(double re, double im) {
	return 0.5 * log1p(expm1((im * (-2.0 * cos(re)))));
}

Java

public static double code(double re, double im) {
	return 0.5 * Math.log1p(Math.expm1((im * (-2.0 * Math.cos(re)))));
}

Python

def code(re, im):
	return 0.5 * math.log1p(math.expm1((im * (-2.0 * math.cos(re)))))

Julia

function code(re, im)
	return Float64(0.5 * log1p(expm1(Float64(im * Float64(-2.0 * cos(re))))))
end

Wolfram

code[re_, im_] := N[(0.5 * N[Log[1 + N[(Exp[N[(im * N[(-2.0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

   2. neg-sub0 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

   3. remove-double-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

   4. remove-double-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

   5. sub0-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

   6. distribute-neg-in 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

   7. +-commutative 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

   8. sub-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

   9. cos-neg 53.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

   10. associate-*l* 53.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

   11. distribute-rgt-neg-in 53.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

   12. *-commutative 53.7%

       \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

3. Simplified 53.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 52.9%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]
6. Step-by-step derivation

   1. log1p-expm1-u 98.7%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)} \]

   2. *-commutative 98.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(im \cdot -2\right)} \cdot \cos re\right)\right) \]

   3. associate-*l* 98.7%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(-2 \cdot \cos re\right)}\right)\right) \]

7. Applied egg-rr 98.7%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right)} \]

8. Final simplification 98.7%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right) \]
9. Add Preprocessing

Alternative 2: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.104:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.104)
   (* 0.5 (* (cos re) (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))
   (if (<= im 5.6e+102)
     (* 0.5 (- (exp (- im)) (exp im)))
     (* (cos re) (* (pow im 3.0) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.104) {
		tmp = 0.5 * (cos(re) * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0))));
	} else if (im <= 5.6e+102) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = cos(re) * (pow(im, 3.0) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.104d0) then
        tmp = 0.5d0 * (cos(re) * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0))))
    else if (im <= 5.6d+102) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = cos(re) * ((im ** 3.0d0) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.104) {
		tmp = 0.5 * (Math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0))));
	} else if (im <= 5.6e+102) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 3.0) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.104:
		tmp = 0.5 * (math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0))))
	elif im <= 5.6e+102:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = math.cos(re) * (math.pow(im, 3.0) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.104)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0)))));
	elseif (im <= 5.6e+102)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(cos(re) * Float64((im ^ 3.0) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.104)
		tmp = 0.5 * (cos(re) * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0))));
	elseif (im <= 5.6e+102)
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = cos(re) * ((im ^ 3.0) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.104], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+102], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.103999999999999995

   1. Initial program 38.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.6%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.6%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.6%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 90.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   if 0.103999999999999995 < im < 5.60000000000000037e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 81.8%

      \[\leadsto 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \color{blue}{1}\right) \]

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.104:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 460.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 5.6e+102)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* (cos re) (* (pow im 3.0) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 5.6e+102) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = cos(re) * (pow(im, 3.0) * -0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 5.6e+102) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 3.0) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 460.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 5.6e+102:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = math.cos(re) * (math.pow(im, 3.0) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 460.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 5.6e+102)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(cos(re) * Float64((im ^ 3.0) * -0.16666666666666666));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 460.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+102], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 460

   1. Initial program 38.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.6%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.6%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.6%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 68.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]

   if 460 < im < 5.60000000000000037e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 3.6%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. log1p-expm1-u 96.1%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)} \]

      2. *-commutative 96.1%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(im \cdot -2\right)} \cdot \cos re\right)\right) \]

      3. associate-*l* 96.1%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(-2 \cdot \cos re\right)}\right)\right) \]

   7. Applied egg-rr 96.1%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right)} \]

   8. Taylor expanded in re around 0 77.9%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 77.9%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-2 \cdot im\right)}\right) \]

   10. Simplified 77.9%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-2 \cdot im\right)}\right) \]

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00155:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00155)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 5.6e+102)
     (* 0.5 (- (exp (- im)) (exp im)))
     (* (cos re) (* (pow im 3.0) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00155) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 5.6e+102) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = cos(re) * (pow(im, 3.0) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00155d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 5.6d+102) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = cos(re) * ((im ** 3.0d0) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00155) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 5.6e+102) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 3.0) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00155:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 5.6e+102:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = math.cos(re) * (math.pow(im, 3.0) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00155)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 5.6e+102)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(cos(re) * Float64((im ^ 3.0) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00155)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 5.6e+102)
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = cos(re) * ((im ^ 3.0) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00155], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+102], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00154999999999999995

   1. Initial program 38.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 38.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 38.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 38.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 38.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.3%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.3%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.3%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 68.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]

   if 0.00154999999999999995 < im < 5.60000000000000037e102

   1. Initial program 99.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 99.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 99.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 99.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 99.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 99.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 99.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 99.7%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 99.7%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 99.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 99.7%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 79.5%

      \[\leadsto 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \color{blue}{1}\right) \]

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto \cos re \cdot \color{blue}{\left({im}^{3} \cdot -0.16666666666666666\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00155:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 410.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (log1p (expm1 (* im -2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 410.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 410.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 410.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 410

   1. Initial program 38.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.6%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.6%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.6%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 68.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]

   if 410 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 5.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)} \]

      2. *-commutative 98.6%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(im \cdot -2\right)} \cdot \cos re\right)\right) \]

      3. associate-*l* 98.6%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(-2 \cdot \cos re\right)}\right)\right) \]

   7. Applied egg-rr 98.6%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right)} \]

   8. Taylor expanded in re around 0 74.8%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-def 74.8%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-2 \cdot im\right)}\right) \]

   10. Simplified 74.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-2 \cdot im\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.2e-7)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.2e-7) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.2d-7) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.2e-7) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.2e-7:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.2e-7)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.2e-7)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.2e-7], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.2000000000000001e-7

   1. Initial program 37.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 37.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 37.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 37.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 37.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 37.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 37.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 37.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 37.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 37.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 37.9%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 37.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 37.9%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 37.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 68.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]

   if 3.2000000000000001e-7 < im

   1. Initial program 99.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 99.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 99.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 99.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 99.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 99.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 99.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 99.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 99.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 99.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 99.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 99.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 68.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   6. Taylor expanded in re around 0 50.4%

      \[\leadsto 0.5 \cdot \left(\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right) \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.9 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.9e+55)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* (pow im 3.0) -0.16666666666666666)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.9e+55) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.9d+55) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.9e+55) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.9e+55:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.9e+55)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.9e+55)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = (im ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.9e+55], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.9e55

   1. Initial program 41.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 41.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 41.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 41.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 41.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 41.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 41.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 41.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 41.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 41.3%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 41.3%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 41.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 41.3%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 41.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 65.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]

   if 1.9e55 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 77.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 77.5%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 56.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   8. Step-by-step derivation

      1. *-commutative 56.9%

         \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} \]

   9. Simplified 56.9%

      \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.9 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.4:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.4) (* 0.5 (* im -2.0)) (* (pow im 3.0) -0.16666666666666666)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.4) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.4d0) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else
        tmp = (im ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.4) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = Math.pow(im, 3.0) * -0.16666666666666666;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.4:
		tmp = 0.5 * (im * -2.0)
	else:
		tmp = math.pow(im, 3.0) * -0.16666666666666666
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.4)
		tmp = Float64(0.5 * Float64(im * -2.0));
	else
		tmp = Float64((im ^ 3.0) * -0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.4)
		tmp = 0.5 * (im * -2.0);
	else
		tmp = (im ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.4], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.39999999999999991

   1. Initial program 38.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.6%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.6%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.6%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.6%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 68.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]

   6. Taylor expanded in re around 0 42.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im\right)} \]

   if 2.39999999999999991 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

      3. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

      4. remove-double-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 67.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 67.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 49.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
   8. Step-by-step derivation

      1. *-commutative 49.2%

         \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} \]

   9. Simplified 49.2%

      \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.4:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.0% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot -2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im -2.0)))

C

double code(double re, double im) {
	return 0.5 * (im * -2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (im * (-2.0d0))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * -2.0);
}

Python

def code(re, im):
	return 0.5 * (im * -2.0)

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * -2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * -2.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

   2. neg-sub0 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

   3. remove-double-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

   4. remove-double-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

   5. sub0-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

   6. distribute-neg-in 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

   7. +-commutative 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

   8. sub-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

   9. cos-neg 53.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

   10. associate-*l* 53.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

   11. distribute-rgt-neg-in 53.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

   12. *-commutative 53.7%

       \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

3. Simplified 53.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 52.9%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im\right)} \cdot \cos re\right) \]

6. Taylor expanded in re around 0 32.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im\right)} \]

7. Final simplification 32.9%

   \[\leadsto 0.5 \cdot \left(im \cdot -2\right) \]
8. Add Preprocessing

Alternative 10: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

double code(double re, double im) {
	return -1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.0d0
end function

Java

public static double code(double re, double im) {
	return -1.0;
}

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0;
end

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 53.7%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

   2. neg-sub0 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} + \left(-e^{im}\right)\right) \]

   3. remove-double-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-\left(-e^{-im}\right)\right)} + \left(-e^{im}\right)\right) \]

   4. remove-double-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{-\left(-im\right)}}\right)\right) \]

   5. sub0-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

   6. distribute-neg-in 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

   7. +-commutative 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

   8. sub-neg 53.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

   9. cos-neg 53.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

   10. associate-*l* 53.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

   11. distribute-rgt-neg-in 53.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

   12. *-commutative 53.7%

       \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

3. Simplified 53.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
4. Add Preprocessing

6. Applied egg-rr 2.9%

   \[\leadsto 0.5 \cdot \left(\color{blue}{-2} \cdot \cos re\right) \]

7. Taylor expanded in re around 0 2.9%

   \[\leadsto 0.5 \cdot \left(-2 \cdot \color{blue}{1}\right) \]

8. Final simplification 2.9%

   \[\leadsto -1 \]
9. Add Preprocessing

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024019 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))